A couple of ways to get a quick feel for the business the company is in are go ask the Salesman what they are selling. And go ask the customer service representatives what kinds of questions they are getting. Try not to ask the managers, ask the foot soldiers.
If you can, tell us what industry your in.
There are a variety of choices for learning about how to go about Business Analysis and/or what a BA does.
But the first thing we are supposed to do is ask questions in plain english and answer in plain english. If you do an interview make sure you take notes at the interview and then do a "data dump" right after the interview to produce a good description of what you think you learned. Then, if possible, run the "report" by the interviewee and see if you really heard what they were saying.
The companys top managers probably have more than one idea about what is working and what is not working with the Website and/or Company. With your Managers permission ask for a 15 minute interview with some of them. Make sure you have prepared a list of questions and send them ahead of the meeting.
There is a user requirements template for word on this website. It might help. I also like the book "7 Steps to mastering Business Analysis" by Ms. Carkenord
Since I posted this message I have finished my first read of this book. It provides a good overview with sufficient details to allow a newby like me to begin to get some idea of just how complicated being a BA can be. For a new BA, I can recommend it un-reservedly because it makes a systematic attempt to pull together what a BA does in the waterfall and interative environments. There are a few nod's to Agile development in there too.
HTH's
Tom M